Graphing Summary

Trigonometry · Axiom Academy

SUMMARY Graphing Trigonometric Functions Trigonometric graphs reveal the periodic nature of circular functions. Understanding transformations allows modeling of any periodic phenomenon from heartbeats to planetary orbits. Sine Function: y = sin(x) oscillates smoothly between -1 and 1, starting at the origin (0,0) and completing one cycle every 2π radians Cosine Function: y = cos(x) has the same wave shape as sine but starts at its maximum (0,1), shifted π/2 to the left Key Points: Both functions have zeros, maxima, and minima at regular intervals. Sine: zeros at nπ ; Cosine: zeros at π/2 + nπ Relationship: cos(x) = sin(x + π/2) , showing cosine is a phase-shifted sine wave Amplitude: Maximum displacement from the midline. For y = A sin(x) , amplitude is |A| . Graph oscillates between -|A| and |A| Period: Length of one complete cycle. For y = sin(Bx) , period is 2π/|B| . Larger B means faster oscillation Frequency: Number of cycles per unit interval. Frequency = |B|/2π , inversely related to period Midline: The horizontal center line of oscillation. For standard functions it's y = 0 , but can shift vertically Graphing a Transformed Function: y = 3sin(2(x - π/4)) + 1 Identify amplitude: A = 3 , so the graph oscillates 3 units above and below the midline Find the period: B = 2 , so Period = 2π/2 = π . The wave completes one full cycle every π radians Determine phase shift: C = π/4 , so the entire graph shifts π/4 units to the right

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